We pursue two goals in this paper. First, we want to extend the theory of the spaces \(B^{\lbrace s_{k}\rbrace }_{p,q}(\mathbb {R}^{n})\)  with \(p,q \in (0,\infty ]\) 
to a much wider class of weight sequences.
In doing so, the approach developed by Rychkov [1]} will be used instead of the Fourier analysis.
The class of weighted sequences to be introduced below turns out to be so broad that it will envelope both the weighted Besov spaces of [1]} and the Besov spaces of variable smoothness
(only for constant \(p\)  and \(q\) )
which were actively studied by H. Kempka with coauthors in [3]}, [4]} (for details see Remark REF  below).
It is also worth noting that Besov spaces of variable smoothness are first dealt with in this degree of generality.
